Mutation in triangulated categories and rigid Cohen–Macaulay modules

By Osamu Iyama, Yu Yoshino. 2006

572 citations recorded by the source corpus.

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/22465487bb8b747c52e655c166ee354ba9099f3d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The periodicity conjecture for pairs of Dynkin diagrams (2010, background): “By the construction of T ∗ 1 in [40], we have a triangle T ∗ 1 i // E p // T1 ε // ΣT ∗ 1 , where p is a minimal right add(T ′)-approximation of T1.”
Silting mutation in triangulated categories (2010, background): “Two important cases are well known: One is ‘cluster tilting mutation’ [11, 23, 24] for cluster tilting objects which is applied for categorification of Fomin–Zelevinsky cluster algebras, and the other is ‘exceptional mutation’ for exceptional sequences, which is used to study the structure of derive”
On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories (2007, unknown): “with B and B0 belonging to T \ T 0, cf. e.g. [3] [12] [ 15 ].”
Cluster tilting for one-dimensional hypersurface singularities (2007, methodology): “Since C is 2-Calabi-Yau, G contains F .B y [ Am , 2.2.1], G is generated by a single element g. By the condition F ∈� g� ,w e have the above”
Mutation of cluster-tilting objects and potentials (2008, methodology): “Remark The definition given here is more general than 2-almost split sequences and AR 4-angles in [I2, IY ].”
Cluster tilting for higher Auslander algebras (2008, background): “Fix any indecomposable object X ∈ D. Since Un(�) is closed under S ±1 n , the following conditions are equivalent (e.g. [ IY , Prop. 3.5]).”
Coloured quiver mutation for higher cluster categories (2008, methodology): “Assume m ≥ 2. For the first claim see [ IY ], while the second claim then follows from Lemma 3.2(e).”
Triangulated quotient categories revisited (2016, methodology): “ies carry triangulated structures (see Theorem 3.15). As applications, our result uniﬁes the triangulated quotient category construction considered by Happel [Ha], Beligiannis [B1, B2], Iyama-Yoshino [IY], Jørgensen [J], and Liu-Zhu [LZ]. We note that there are some other constructions of quotients ”
On two examples by Iyama and Yoshino (2008, background): “ 9 6. The Iyama-Yoshino examples (again) 12 7. A remark on gradability of rigid modules 14 Appendix A. Generators of singularity categories 15 References 21 1. Introduction Throughout k is a ﬁeld. In [17] Iyama and Yoshino consider the following two settings. Example1.1. Let S = k[[x 1,x 2,x 3]] and”